John Nash: game theory

On the Edge: The Art of Risking Everything

by Nate Silver  · 12 Aug 2024  · 848pp  · 227,015 words

focuses on man vs. machine and the advent of computer solvers, which have revolutionized poker. The basis for these solvers is game theory, which I discuss at length—alongside expected value, game theory is one of the most important concepts in the River. However, in chapter 2, Perception, we learn that some of

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these programs are trained by essentially playing against themselves. And they’re designed to achieve a Nash equilibrium or game-theory optimal (GTO) style of play. “Nash equilibrium” is named after the American mathematician John Nash, a discovery for which he shared the Nobel Prize. (Nash is also famous as a result of his

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portrayal by Russell Crowe in the movie A Beautiful Mind.) I’ll have a lot more to say about game theory later in this chapter, but the

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like Dominik Nitsche and Christoph Vogelsang. Somewhat befitting the cultural stereotype, German players are known for highly precise play, and they were early adopters of game theory and computer solutions. These players were using technical terms, like “blockers” and “combos,”[*5] that Negreanu thought were gibberish. “Then I realized that playing

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the 2022 Super High Roller Bowl for $3.3 million. But where do these new strategies come from, exactly? It’s time for some game theory. The Mastermind of Game Theory “Genius” might be an overused term, but it’s the only appropriate label for John von Neumann. Born in Hungary, where he was

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our best life, but they are too. What’s the equilibrium that emerges when everyone pursues their best strategy? That’s what game theory is all about. I find game theory appealing because like other people in the River, I often find myself in highly competitive settings. How should you play your cards if

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poker players do today. Not everything lends itself as well to algorithmic optimization as poker does—but the warp-speed advancements in poker demonstrate that game theory often translates reasonably well into practice. The High-Stakes Game of GTO vs. Exploitative Play Exploit and you risk being exploited. This may be

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value via arbitrage, “information not reflected in the line such as injuries,” and through clever betting tactics. Top-down is in line with a game-theory equilibrium that assumes everyone playing the sports-betting game is pretty smart, resulting in a market where betting lines are reasonably efficient and there aren

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sells, then Satoshi definitely wants to sell too. Otherwise he’ll be the one left holding the bag and his ₦ will become worthless. The Game Theory of a Shitcoin Bubble Satoshi Sells Satoshi Holds Pepe Sells Sale order is determined randomly. One of them makes a $100 profit and the other

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, especially in its strictest forms, is actually relatively unpopular among philosophers. Kevin Zollman, a Carnegie Mellon philosopher who has also written books on how game theory can be applied to everyday life, told me that he thinks utilitarianism can be seductive to quantitatively inclined people because of its promise of mathematical

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curve. But they gradually infer higher-level concepts. They may notice, for instance, that large bets usually signify either very strong hands or bluffs, as game theory dictates. These days, most players will also study with computer solvers, going back and forth between inductive reasoning (imputing theory from practice) and deductive

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the undistilled version of it is often dangerous. Finally, there is reciprocity. This is the most Riverian principle of all, since it flows directly from game theory. Treat other people as intelligent and capable of reasonable strategic behavior. The world is dynamic, and although people may not be strictly rational, they’re

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a book about hypercompetitive gamblers and then say something like “turn the other cheek.” Reciprocity sometimes does mean reciprocating. Deterrence plays a big role in game theory. Sometimes you need to stand your ground. Nonetheless, we ought to give other people the benefit of the doubt more often. Respect and trust

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Agency: As defined more completely in chapter ∞, being empowered to make robust, well-informed decisions; knowing which factors are inside one’s control. Agent: In game theory or AI, an entity possessed of enough intelligence to make reasonable strategic choices. AGI: Artificial general intelligence. The term lacks a clear definition but refers

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principles to deduce conclusions about specific cases, e.g., using constitutional principles to determine legal practices in particular situations. See also: inductive reasoning. Defect (game theory): To snitch or act self-interestedly rather than cooperate, as predicted by the prisoner’s dilemma. Degen, degenerate: A person who has a tendency to

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by their consequences. Deterministic: The opposite of probabilistic: an outcome that is preordained or strictly predictable with probability of exactly 1 or 0. Deterrence (game theory): Preventing aggression from your opponent through the threat of escalation, e.g., by credibly threatening to go all-in in poker or launch a retaliatory

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costs to their present value. A higher discount rate implies a shorter time horizon. Domain knowledge: Expertise in a particular subfield. Dominant strategy: In game theory, a move that is always the best play no matter what your opponent does; the opposite is a dominated strategy. DonBest: The most well-reputed

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s knowledge and the ability to acknowledge uncertainty in understanding the truth, rooted in the study of epistemology, the philosophy of knowledge acquisition. Equilibrium: See: game-theory equilibrium. Equity (poker): Your share of the pot in expected-value terms, e.g., a player with a flush draw after the flop has

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’re facing a tractable problem with a quantifiable answer. Exploitative strategy: An approach designed to take advantage of an opponent who isn’t playing a game-theory optimal strategy. Be careful, because exploitative strategies can be counterexploited and open you up to EV loss. Externality: A cost imposed on others that

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carry considerable weight in any model of the world; the loss of verifiable ground truth is one hazard of a more virtual world. GTO: See: game-theory optimal. Hallucination (AI): Often “creative” false information produced by LLMs when they don’t know the answer but pretend they do. Handicapper: Someone who

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won’t attack one another because they’d be assured of a devastating retaliatory strike if they did. Nash equilibrium: After the Princeton mathematician John Nash, a game theory solution in which all participants have optimized their EV and there are no further gains from unilateral changes in strategy. Nash proved that all games

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participant’s gain is exactly balanced by another’s loss, with the total utility remaining constant. Although zero-sum games were the original basis for game theory, many real-world scenarios like nuclear deterrence involve mixed motives blending elements of competition and cooperation. notes Chapter 0: Introduction Seminole Hard Rock: David

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GO TO NOTE REFERENCE IN TEXT Kant’s Golden Rule: Janet Chen, Su-I Lu, and Dan Vekhter, “Applications of Game Theory,” Game Theory, cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/applications.html. GO TO NOTE REFERENCE IN TEXT a nightly profit: That is, 2,000 slices times a $1.50

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“ ‘The Pandemic Is a Prisoner’s Dilemma Game,’ ” The New York Times, December 20, 2020, sec. Health, nytimes.com/2020/12/20/health/virus-vaccine-game-theory.html. GO TO NOTE REFERENCE IN TEXT when OPEC colludes to set: “OPEC (Cartel),” Energy Education, energyeducation.ca/encyclopedia/OPEC_(cartel). GO TO NOTE REFERENCE

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NOTE REFERENCE IN TEXT analogized to the prisoner’s dilemma: Janet Chen, Su-I Lu, and Dan Vekhter, “Applications of Game Theory,” Game Theory, cs.stanford.edu/people/eroberts/courses/soco/projects/1998-99/game-theory/applications.html. GO TO NOTE REFERENCE IN TEXT called “rule utilitarianism”: Stephen Nathanson, “Utilitarianism, Act and Rule,” Internet Encyclopedia

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bias, 481 Collison, Patrick, 262, 263, 466, 494 combo (poker), 481 community cards (board) (poker), 41, 480, 481 competitiveness, 25–26 abuse and, 118 game theory and, 52, 66 lack of money drive and, 243 poker and, 112, 118, 120, 243 resentment and, 223, 277–78 River-Village conflict and, 28

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311–18, 317 business model of, 308–9 COVID-19 and, 6, 310, 312–13 defined, 482 focal points and, 329 fraud and, 124, 308 game theory and, 316–18, 317, 327 gender and, 312 history of, 322–24, 326 HODL, 308, 309, 310, 312, 317, 318, 487 meme creation of

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value, 60, 482 decision science, 427–28 decoupling, 24–25, 26, 27, 352, 482, 505n deductive reasoning, 482 Deeb, Shaun, 95 Deep Blue, 60 defect (game theory), 482 degens (gambling), 9, 114, 278–79, 482 Demis Hassabis, 416 Denton, Nick, 249–50n, 274 deontology, 359, 368, 481, 482 Designing Casinos to

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returns (finance), 484 existential risk, 401–3 biotechnology and, 457n defined, 484 definitions of, 442–44 effective altruism/rationalism and, 355, 380 futurism and, 380 game theory and, 421–22 hedonism and, 376–77 nanotechnology and, 457n von Neumann on, 419–20 See also AI existential risk; nuclear existential risk expected value

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477 secular stagnation and, 466 whales, 83–84, 139, 151, 156–57, 187, 207–8, 500 See also casinos; poker Game of Gold, 100–101 game theory AI and, 47 alpha and, 242 blockers, 229 bluffing and, 64, 70–75, 101 competitiveness and, 52, 66 cryptocurrency and, 316–18, 317, 327 deception

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486 Graham, Paul, 405, 406, 413, 539n grand-world problem, 486 Great Man Theory, 344 Greenberg, Spencer, 400 ground truth, 486 group selection, 429n GTO (game theory optimal) strategies (poker), 47, 62, 63, 65–67, 71–72, 485–86, 508n, 509n See also EV maximizing; Nash equilibrium Gurley, Bill, 259, 269, 

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–61, 366–67, 368, 377, 487, 533n, 538n independence, 25, 31, 239–40, 249, 268, 273, 358 See also contrarianism index funds, 487 indifferent (game theory), 487 inductive reasoning, 487 Industrial Revolution, 461–62, 462, 487 infinite ethics, 360, 364–65, 487 inflection points, 487 innovator’s dilemma, 273–74n, 487

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Chris, 12, 43, 68, 493 Monnette, John, 103–4 moral hazard, 30, 261, 490 moral philosophy consequentialism, 359, 481, 533n deontology, 359, 368, 481, 482 game theory and, 367–68 impartiality, 358–59, 360–61, 366–67, 368, 377, 487, 533n, 538n modern value proposal, 469–72 moral parliament, 364, 470 overfitting

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NOT INVESTMENT ADVICE, 491 Noyce, Robert, 257 NPC (nonplayer character) syndrome, 378–79, 490 nuclear existential risk, 407, 420–30 Bayesian reasoning on, 423 game theory and, 58, 328, 420–21, 424, 426, 483 Kelly criterion and, 408–9 mutually assured destruction and, 58, 421, 424–27, 488, 490 nuclear proliferation

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482 edge and, 22, 63, 86 effective altruism and, 347–48, 367 estimation ability and, 237–38 fictional portrayals of, 45, 112, 134, 333, 487 game theory development and, 22, 50–51 game trees in, 61, 508n Garrett-Robbi hand, 80–86, 89, 117, 123–29, 130, 444–45, 512n gender

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, 250, 456–57, 472 solvers (poker), 62 bluffing and, 74, 75, 78, 509n Doyle Brunson and, 43 defined, 497 exploitative strategies and, 78–79 game theory and, 22, 60–61, 62–65, 71, 74 Garrett-Robbi hand and, 125 invention of, 60–61 optionality and, 76–77 space travel, 218–19

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cheating, 136 closing line value, 205–6, 206, 481 computer applications and, 172 contrarianism and, 240 deception and, 206–7 fantasy sports, 198, 199, 483 game theory and, 171 getting (money) down, 48, 204–9 inside information and, 177, 187n, 194, 197n, 212–13 Kelly criterion and, 397, 399 key skills

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Kip, 351 Vogelsang, Christoph, 49 von Neumann, John autism and, 283 on computer applications, 61 Dr. Strangelove and, 425n existential risk and, 419–20, 422 game theory development and, 22, 50–51 Kelly criterion and, 396 on technological singularities, 450n Voulgaris, Haralabos “Bob,” 192–94, 204, 205, 207n, 235–36, 518n W

Prisoner's Dilemma: John Von Neumann, Game Theory, and the Puzzle of the Bomb

by William Poundstone  · 2 Jan 1993  · 323pp  · 100,772 words

NEUMANN The Child Prodigy Kun’s Hungary Early Career The Institute Klara Personality The Sturm und Drang Period The Best Brain in the World 3 GAME THEORY Kriegspiel Who Was First? Theory of Games and Economic Behavior Cake Division Rational Players Games as Trees Games as Tables Zero-Sum Games Minimax

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N-Person Games 4 THE BOMB Von Neumann at Los Alamos Game Theory in Wartime Bertrand Russell World Government Operation Crossroads The Computer Preventive War 5 THE RAND CORPORATION History Thinking About the Unthinkable Surfing, Semantics, Finnish Phonology Von Neumann at RAND John Nash The Monday-Morning Quarterback 6 PRISONER’S DILEMMA The Buick

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Francis Matthews Aftermath Public Reaction Was It a Trial Balloon? The MacArthur Speech Orvil Anderson Press Reaction How Many Bombs? Coda 8 GAME THEORY AND ITS DISCONTENTS Criticism of Game Theory Utility and Machiavelli Are People Rational? The Ohio State Studies 9 VON NEUMANN’S LAST YEARS The H-Bomb A Very Fine Tiger

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TIT FOR TAT in the Real World 13 THE DOLLAR AUCTION Escalation Shubik’s Dollar Auction Dollar Auctions in Real Life Strategies Rational Bidding Where Game Theory Fails The Largest-Number Game Feather in a Vacuum Bibliography About the Author Copyright 1 DILEMMAS A man was crossing a river with his

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rational) humans. Most great advances in science come when a person of insight recognizes common elements in seemingly unrelated contexts. This describes the genesis of game theory. Von Neumann recognized that parlor games pose elemental conflicts. It was these conflicts, normally obscured by the window dressing of cards and chessmen and dice

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qualify as a clever contribution to recreational mathematics. Von Neumann saw more profound implications. He intended the minimax theorem to be the cornerstone of a game theory that would eventually encompass other types of games, including those of more than two players and those where the players’ interests partly overlap. So expanded

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sciences. At the same time he insists that the best mathematics is usually inspired by practical problems. This can be read as a defense of game theory (among other things) to those fellow mathematicians who deprecated it as an applied field. Von Neumann warns that “pure” mathematics … becomes more and more

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form it was a popular lunch-hour pastime at the RAND Corporation, and von Neumann played the game on visits there. To some critics, game theory is the twentieth century’s Kriegspiel, a mirror in which military strategists see reflected their own preconceptions. The comparison is revealing even while being unfair

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carry out activities which have no immediate goal (other animals play only while young) in order to prepare himself for long-term strategies and plans.” Game theory is not about “playing” as usually understood. It is about conflict among rational but distrusting beings. Von Neumann escaped revolution and terrorism in Hungary and

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one of repeated conflict. In his letters to his wife Johnny talks of double-crossing, reprisals, and boundless distrust. That’s part of what game theory is about. Game theory was the brainchild of a cynic. Some commentators have suggested that von Neumann’s personal cynicism influenced the theory. It is conceivable that von

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… I never learned the subject and never learned to like it.” WHO WAS FIRST? Von Neumann cannot be given undivided credit for the invention of game theory. Beginning in 1921, seven years before von Neumann’s first paper, French mathematician Emile Borel published several papers on “la théorie du jeu.” The

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and took up the problem of bluffing, just as von Neumann would. Borel appreciated the potential economic and military applications of game theory. Indeed, Borel warned against overly simplistic applications of game theory to warfare. He was not talking off the top of his head. Borel, who had held public office, became minister

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Borel had not) the famous “minimax theorem.” This important result immediately gave the field mathematical respectability. THEORY OF GAMES AND ECONOMIC BEHAVIOR Von Neumann wanted game theory to reach a larger audience than mathematicians. He felt the developing field would be of most use to economists. He teamed with an Austrian economist

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In a game of ticktacktoe with a child, you might even play to lose. This is all well and fine. It is not what game theory is about. Game theory is about perfectly logical players interested only in winning. When you credit your opponent(s) with both rationality and a desire to win, and

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has the maximum minimum. If the second player had to go first, he would want the minimum maximum. Again this is the random strategy. Game theory suggests the lower right cell as the natural outcome. Both players should choose randomly. Once again we find an equilibrium between the players’ opposed interests

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to begin with, but players form opinions about their opponent’s hands from their bids. Judicious bluffing prevents a player from being too predictable. Game theory has important analogies in biology. A person who inherits the relatively rare sickle-cell anemia gene from one parent has greater immunity to malaria, but

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win as much money as possible or as genes mindlessly reproducing as much as natural selection permits. We’ll hear more about biological interpretations of game theory later. THE MINIMAX THEOREM The minimax theorem proves that every finite, two-person, zero-sum game has a rational solution in the form of

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is a rational solution in that both parties can convince themselves that they cannot expect to do any better, given the nature of the conflict. Game theory’s prescriptions are conservative ones. They are the best a rational player can expect when playing against another rational player. They do not guarantee

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of the fast-growing use of automata.” And though von Neumann himself might not have imagined it, computers would become important for basic research in game theory. Robert Axelrod’s studies on iterated prisoner’s dilemmas would have scarcely been feasible without computers. Meanwhile, Klara became one of the first computer programmers

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1940s and early 1950s, few of the biggest names of game theory and allied fields didn’t work for RAND, either full-time or as consultants. Besides von Neumann, RAND employed Kenneth Arrow, George Dantzig, Melvin Dresher, Merrill Flood, R. Duncan Luce, John Nash, Anatol Rapoport, Lloyd Shapley, and Martin Shubik—nearly all

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which talent was concentrated so exclusively at one institution. Most of the above workers had left by 1960, but the RAND diaspora continued to dominate game theory throughout the academic world. VON NEUMANN AT RAND John von Neumann’s formal association with the RAND Corporation began in 1948. On December 16,

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proof of his feelings. No one this side of Elysium, he says, can demonstrate love all the time without being boring. JOHN NASH After von Neumann, the next major figure in game theory was another RAND consultant, John F. Nash. Nash was born in Bluefield, West Virginia, in 1928. He studied mathematics at

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by juggling a bicoastal career as RAND consultant and professor at the Massachusetts Institute of Technology. In the late 1940s and early 1950s, Nash extended game theory in a direction von Neumann and Morgenstern had not taken it. Nash studied “noncooperative” games where coalitions are forbidden. Von Neumann and Morgenstern’s

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, you have to assume that all four outcomes are possible. The more interesting part of the argument applies to the perfectly rational players postulated by game theory. Presumably there is only one “rational” way to act in a prisoner’s dilemma. Therefore only the two mutual outcomes (both cooperate or both

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that more fairly represent minority interests. John Nash became increasingly paranoid. He would pester his colleagues with peculiar ideas for tightening security at RAND. He was eventually committed to a psychiatric hospital for treatment. He recovered and joined the Institute for Advanced Study. CRITICISM OF GAME THEORY Views on game theory were changing. A decade after

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the publication of Theory of Games and Economic Behavior, there was a correction to the early euphoria. Game theory was deprecated, distrusted, even reviled. To many, game theory, ever intertwined with the figure of John von Neumann, appeared to encapsulate a callous cynicism about the fate of the human race

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Misgivings continued well into the 1980s and perhaps to the present day. Steve J. Heims, in his John Von Neumann and Norbert Wiener (1980) wrote: “Game theory portrays a world of people relentlessly and ruthlessly but with intelligence and calculation pursuing what each perceives to be his own interest.... The harshness of

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with no ethical scruples about betrayal or psychic rewards for benevolence to complicate the picture. The mistake is in thinking that game theory is specifically about such people. Like arithmetic, game theory is an abstraction that applies to the real world only to the extent that its rigorous requirements are met. Consider the person

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wrong? Of course not. If you count wrong, you can’t blame arithmetic. Likewise, assessment of utilities is a prerequisite for applications of game theory. Here arithmetic and game theory diverge. Any two people who accurately count a group of pennies must come to the same result. Utility, though, is subjective by definition.

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people are likely to rank a set of game outcomes differently, provided the outcomes are not cash prizes but very complex states of human affairs. Game theory is a kaleidoscope that can only reflect the value systems of those who apply it. If game theoretic prescriptions sometimes seem Machiavellian, it is

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generally because the value systems of those who applied the game theory are Machiavellian. There is also a crucial difference between zero-sum games with a saddle point and those without. A saddle point exists even

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peace, limited war, and nuclear holocaust. One can pull “reasonable” numbers out of the air, to be sure, but that defeats the purpose of applying game theory, which was to provide recommendations more accurate than intuition. As Rapoport noted in Scientific American: Unless this more precise quantification of preferences can be made

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a set of studies done at the RAND Corporation in 1952 and 1954. The research team, which included John Nash, tried to establish or refute the applicability of von Neumann’s n-person game theory. In the RAND experiments, four to seven people sat around a table. They played a “game” mimicking the

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comments on those who play the geopolitical version of it. Incidentally, the game Russell describes is now considered the “canonical” chicken, at least in game theory, rather than the off-the-cliff version of the movie: Since the nuclear stalemate became apparent, the Governments of East and West have adopted the

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Herman Kahn’s On Thermonuclear War (1960) credits Russell as the source of the chicken analogy. Chicken readily translates into an abstract game. Strictly speaking, game theory’s chicken dilemma occurs at the last possible moment of a game of highway chicken. Each driver has calculated his reaction time and his car

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, people act to maximize those numerical things. Hence the analogy to numerical maximization. Where people don’t faithfully maximize individual gain, the analogy fails. Game theory has nothing to say. A rational person who consistently foregoes the gains of defection when such actions can’t influence the other player’s choice

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fill now. Natural selection “chooses” or “prefers” the behaviors that will maximize survival value. This is all we need to apply the mathematics of game theory, even though no conscious choices or preferences may be involved. Those animals that get the highest “scores” are most likely to survive and reproduce. The

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Journal of Conflict Resolution and later in Axelrod’s book The Evolution of Cooperation (1984) this work ranks among the most significant discoveries of game theory. Axelrod came to game theory by a circuitous route typical of the field. He was a math major at the University of Chicago, where he took Morton Kaplan

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They were more likely to cooperate with familiar bats that they could expect to interact with in the future. COOPERATION AND CIVILIZATION We all use game theory as unconsciously as sticklebacks or bats. A human society is a group interacting repeatedly. Some interactions pose choices between self and group interest. How often

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players, in pursuing their own ends, will be forced into a socially undesirable position.” Human history is not one of ever-increasing cooperation, though. Game theory may help to understand that, too. TIT FOR TAT is not the only conditional strategy that is evolutionarily stable or nearly so. Once entrenched, other

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-USSR conflict will probably lead to an armed “total” collision, and that a maximum rate of armament is therefore imperative. Von Neumann may have underestimated game theory’s ability to represent “neurotic” irrational acts. The “dollar auction” is a game of escalation and outrageous behavior. It may appear to be the

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again. Then quit. No use being a damn fool about it.” Shubik’s dollar auction demonstrates the difficulty of using von Neumann and Morgenstern’s game theory in certain situations. The dollar auction game is conceptually simple and contains no surprise features or hidden information. It ought to be a “textbook

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from navigating rockets to the moon. Small uncertainties about the masses of celestial bodies lead to proportionately small uncertainties about the trajectory of a rocket. Game theory’s solution to the dollar auction takes an unusual form. The rational course of actions depends on knowing with certainty which of two inexact quantities

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definition of rationality, it’s hard to accept that rationality ever runs aground. In a one-shot prisoner’s dilemma, the type of rationality game theory recognizes leads to mutual defection, and all attempts to come up with alternative types of rationality have failed. Real-world dilemmas are built of subjective

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Switch, and Other West African Stories. New York: Henry Holt & Co., 1947. Cousins, Norman. Modern Man Is Obsolete. New York: Viking, 1945. Davis, Morton D. Game Theory: A Nontechnical Introduction. New York: Basic Books, 1970. Davis, Nuel Pharr. Lawrence & Oppenheimer. New York: Simon & Schuster, 1968. Dawkins, Richard. The Selfish Gene. 2d

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In Journal of Conflict Resolution 32 (1988): 457–72. ______. Fights, Games, and Debates. Ann Arbor: University of Michigan Press, 1960. ______. “The Use and Misuse of Game Theory.” In Scientific American (December 1962): 108–14+. Rousseau, Jean Jacques. A Discourse on Inequality. Translated by Maurice Cranston. London: Penguin, 1984. Russell, Bertrand. The Autobiography

A Beautiful Mind

by Sylvia Nasar  · 11 Jun 1998  · 998pp  · 211,235 words

Reynolds, © copyright 1961 by Schroeder Music Co. (ASCAP). Used by permission. All rights reserved. “John F. Nash Jr.” (Autobiographical Essay) and “The Work of John Nash in Game Theory” (Nobel Seminar), in Les Prix Nobel 1994 (Stockholm: Norstedts Tryckeri, 1995). Copyright © The Nobel Foundation, 1994. Excerpts from “Waking in the Blue” from Life

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Howard to tell him that John Nash actually had looked like Russell Crowe in the white T-shirt scene. The movie turned Nash into a celebrity. I was on a flight to Mumbai where I was meeting Amartya Sen, also a Nobel laureate in economics, at a game theory conference. The woman to

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my left had just asked me why I was going to India when the flight attendant came by with an Indian newspaper, and there was a photograph of John Nash, the keynote speaker, on the front page, right

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Nash-Moser theorem,” “Nash blowing-up.”54 When a massive new encyclopedia of economics, The New Palgrave, appeared in 1987, its editors noted that the game theory revolution that had swept through economics “was effected with apparently no new fundamental mathematical theorems beyond those of von Neumann and Nash.”55 Even as

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anything Nash had ever imagined, much less experienced. A revolution was taking place in mathematics and Princeton was the center of the action. Topology. Logic. Game theory. There were not only lectures, colloquia, seminars, classes, and weekly meetings at the institute that Einstein and von Neumann occasionally attended, but there were

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handful of anointed followers. Logic, for some reason, was not highly regarded, despite Church’s towering reputation among early pioneers of computer theory. The game theory clique around Tucker was considered quite déclassé, an anomaly in this ivory tower of pure mathematics. Each clique had its own thoughts about the importance

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asked Davis whether Davis had grown up in a slum. Nash appeared to be interested in almost everything mathematical — topology, algebraic geometry, logic, and game theory — and he seemed to absorb a tremendous amount about each of these during his first year.9 He himself recalled, without elaborating, having “studied mathematics

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use of calculus and advanced statistical methods.40 Von Neumann was critical of these efforts, but they surely prepared the ground for the reception of game theory.41 Economists were actually somewhat standoffish, at least compared to mathematicians, but Morgenstern’s antagonism to the economics profession no doubt contributed to that

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interpret Tucker’s demands for revisions — along with von Neumann’s coldly dismissive reaction — as signs that the department would not accept his work on game theory for a dissertation. However, Tucker, who could be surprisingly forceful, eventually convinced Nash to stick with his original conception — and to make the requested

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agreements. By broadening the theory to include games that involved a mix of cooperation and competition, Nash succeeded in opening the door to applications of game theory to economics, political science, sociology, and, ultimately, evolutionary biology.36 Although Nash used the same strategic form as von Neumann had proposed, his approach

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appreciated. When John Williams, the head of RAND’s mathematics department, wrote a primer on game theory, published as a RAND study, it was illustrated with funny little cartoon figures and full of jokey examples starring John Nash, Alex Mood, Lloyd Shapley, John Milnor, and other members of the math department.45 The

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David Blackwell, Sam Karlin, and Abraham Girschick, and economists Paul Samuelson, Kenneth Arrow, and Herbert Simon.13 Most of the RAND military applications of game theory concerned tactics. Air battles between fighters and bombers were modeled as duels.14 The strategic problem in a duel is one of timing. For each

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problems in the theory of noncooperative games. For all intents and purposes, Nash stopped working in the field in 1950. The dominant thrust of game theory at RAND came from the mathematicians, particularly Shapley, and they were guided less by applications than by the mathematics themselves. During the 1950s Shapley

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allow some other big breakthroughs. The atom bomb, for example, was built before physicists understood the structure of particles. The most significant application of game theory to a military problem grew straight out of the theory of duels and helped shape what was probably RAND’s single most influential strategic study

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After a time, RAND’s sponsors grew less enthusiastic about pure research, less tolerant of idiosyncrasies, and more demanding. Mathematicians got bored and frustrated with game theory. Consultants stopped coming and permanent staffers drifted to universities. Nash never returned after the summer of 1954. Flood left for Columbia University in 1953.

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of the same mind. “Whenever we speak of deterrence, atomic blackmail, the balance of terror … we are evidently deep in game theory,” Thomas Schelling wrote in 1960, “yet formal game theory has contributed little to the clarification of these ideas.”37 14 The Draft Princeton, 1950–51 NEITHER THE PROSPECT of playing military

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, players typically opted to “split the difference.” For the designers of the experiment, however, the results merely cast doubt on the predictive power of game theory and undermined whatever confidence they still had in the subject. Milnor was particularly disillusioned.19 Though he continued at RAND as a consultant for another

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conference, organized by Oskar Morgenstern, and attended by virtually the entire game-theory community, amounted to a celebration of cooperative theory. There was little mention of noncooperative games or bargaining. But John Harsanyi, a Hungarian, Reinhard Selten, a German, and John Nash, dressed in odd mismatched clothing, mostly silent, were all there.14

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Laureate to date but Douglass North (presumably excluded because he is an economic historian, not a mathematical economist), as well as every leading contributor to game theory — Kuhn, Shapley, Shubik, Aumann, Harsanyi, Selten, and so forth — but not Nash.31 In late 1988, Ariel Rubinstein, a recently elected Fellow, was

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of Princeton, New Jersey …”3 The behind-the-scenes saga of John Nash’s Nobel Prize is almost as extraordinary as the fact that the mathematician became a Laureate at all. For years after the idea of a prize for game theory was first considered, even Nash’s most ardent admirers considered the likelihood

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. Lindbeck, who was spending the spring term in Cambridge, oversaw the preparations by telephone. The dozen or so invited speakers represented two generations of leading game-theory researchers, mostly theorists and experimentalists, among them John Harsanyi, Reinhard Selten, Robert Aumann, David Kreps, Ariel Rubinstein, Al Roth, Paul Milgrom, and Eric Maskin.

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we are seeing the realization of the true potential of the revolution launched by von Neumann and Morgenstern.”11 And because most economic applications of game theory use the Nash equilibrium concept, “Nash is the point of departure.”12 The revolution has gone far beyond research journals, experimental laboratories at Caltech

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another game theorist, John McMillan, of the University of California at San Diego, to help evaluate the effect of every proposed rule. According to Milgrom, “Game theory played a central role in the analysis of the rules. Ideas of Nash equilibrium, rationalizability, backward induction, and incomplete information, though rarely named explicitly,

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wished he had won the whole thing because he really needed the money badly. Third, Nash said that he had won for game theory and that he felt that game theory was like string theory, a subject of great intrinsic intellectual interest that the world wishes to imagine can be of some utility.

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1950), pp. 48–49, and later as “Non-Cooperative Games,” Annals of Mathematics (1951), pp. 286–95. See also “Nobel Seminar: The Work of John Nash in Game Theory,” in Les Prix Nobel 1994 (Stockholm: Norstedts Tryckeri, 1995). For a reader-friendly exposition of the Nash equilibrium, see Avinash Dixit and Susan Skeath, Games

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from John Nash to Alex Mood, 11.94. 52. R. Nash, interview, 1.7.96. 53. Confidential source. 54. See, for example, Mikhail Gromov, Partial Differential Relations (New York: Springer-Verlag, 1986); Heisuke Hironaka, “On Nash Blowing Up,” Arithmetic and Geometry II (Boston: Birkauser, 1983), pp. 103–11; P. Ordehook, Game Theory and

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interview. 31. Davis, interview. 32. Interviews with Washnitzer and Kuhn. 33. Washnitzer, interview. 34. Tukey, interview. 35. Kuhn, interview. 36. Calabi, interview. 37. Martin Shubik, “Game Theory at Princeton: A Personal Reminiscence,” Cowles Foundation Preliminary Paper 901019, undated. 38. Interviews with Hausner; Davis; Kuhn; Spencer; Leader; Rogers; Calabi; and John McCarthy, professor

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Economics Association meeting on 1.5.96, Nash traced a lineage from Newton to von Neumann to himself. Nash shared von Neumann’s interest in game theory, quantum mechanics, real algebraic variables, hydrodynamic turbulence, and computer architecture. 6. See, for example, Ulam, “John von Neumann,” op. cit. 7. Norman McRae,

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Press, 1944, 1947, 1953). 2. Both von Neumann and Morgenstern came to the seminar. Albert W. Tucker, interview, 10.94. See also Martin Shubik, “Game Theory and Princeton, 1940–1955: A Personal Reminiscence,” Cowles Foundation Preliminary Paper, undated, p. 3; David Gale, interview, 9.20.95; and Harold Kuhn, interview,

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Other Essays (Mountain Center, Calif.: James & Gordon, 1995). Martin Shubik writes, “Even as a graduate student I was struck by the contrast between cooperative game theory, the seeds of which I regarded as already present in Edgeworth and noncooperative theory which was present in Cournot,” Martin Shubik, Collected Works, forthcoming, p

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55. Arrow, interview. 56. Mood, interview. 57. Best, interview. 58. Harold Shapiro, interview. 59. Mood, interview. 60. Danskin, interview. 61. Ibid. 62. Best, interview. 13: Game Theory at RAND 1. Kenneth Arrow, interview, 6.26.95. 2. M. Dresher and L. S. Shapley, Summary of RAND Research in the Mathematical Theory of

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cit. 5. Thomas C. Schelling, The Strategy of Conflict (Cambridge: Harvard University Press, 1960). 6. Ibid. 7. Arrow, interview. 8. See, for example, Martin Shubik, “Game Theory and Princeton,” op. cit.; William Lucas, “The Fiftieth Anniversary of TGEB,” Games and Economic Behavior, vol. 8. (1995), pp. 264–68; Carl Kaysen, interview, 2

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3.5.96. Also postcards from John Nash to Virginia Nash, 8.1.61 and 8.3.61. 10. Alicia Nash, interview, 8.15.96. 11. Interviews with John Danskin, 10.19.95, and Odette Larde, 12.7.95. 12. O. Larde, interview. 13. “Recent Advances in Game Theory,” Princeton, October 4–6, 1961

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. 14. Reinhard Selten, professor of economics, University of Bonn, interview, 6.27.95. 15. John Harsanyi, interview, 6.27.95. 16. Harold Kuhn, personal communication, 8.97. 17. John Nash, “Le Probleme de Cauchy Pour Les Equations Differentielles d

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the first to point out to the author, a reporter at The New York Times, that Nash might share a Nobel. 33. Nobel Symposium on Game Theory: Rationality and Equilibrium in Strategic Interaction, Bjork-born, Sweden, June 18–20, 1993. 34. Confidential source who attended the conference. 35. Persson, interview. 36

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, interview. 44. Löfgren, interview. 45. Lindbeck, interview. 46. Ibid. 47. Ibid. 48. Shapley’s most important work is in cooperative game theory while Schelling’s work is in applications of game theory. 49. Lindbeck, interview. 50. Ibid. 51. The sketch of Stahl is based on interviews with his brother Ingolf Stahl, 2.12

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conference, “Market Design: Spectrum Auctions and Beyond,” Princeton University, 11.9.95. 6. Peter C. Cramton, “Dealing with Rivals? Allocating Scarce Resources? You Need Game Theory” (Xerox, 1994). Nash provided the fundamental theory used to analyze and predict behavior in simple games in which rational players have complete knowledge of each

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Interaction,” Symposium on Experimental Economics, Econometric Society, Seventh World Congress, August 1995 (draft: September 1994). 9. See, for example, Robert Gibbons, “An Introduction to Applicable Game Theory,” Journal of Economic Perspectives, vol. 11, no. 1 (Winter 1997), pp. 127–49. 10. Avinash Dixit, interview, 7.97. 11. Avinash Dixit, as quoted

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also McMillan, “Selling Spectrum Rights,” op. cit., pp. 153–55. 18. Ibid. 19. See, for example, McMillan, “Selling Spectrum Rights,” op. cit.; Paul Milgrom, “Game Theory and Its Use in the PCS Spectrum Auction,” Games ’95, conference, Jerusalem, 9.29.95. 20. Milgrom, Auction Theory for Privatization, op. cit. 21. Ibid

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, Avinash, and Susan Skeath. Games of Strategy. New York: W. W. Norton, 1997. Eatwell, John, Murray Milgate, and Peter Newman, eds. The New Palgrave: Game Theory: New York: W. W. Norton, 1989. Ewing, John H., ed. A Century of Mathematics. Washington, D.C.: The Mathematical Association of America, 1994. Gardner, Howard

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Introduction, “A Celebration of John F. Nash, Jr.,” Duke Mathematical Journal vol. 81, no. 1 (1995), pp. i-v. ———. “Nobel Seminar: The Work of John Nash in Game Theory, December 8, 1994,” Les Prix Nobel 1994. Stockholm: Norstedts Tryckeri, 1995. Larde, Enrique. The Crown Prince Rudolf: His Mysterious Life After Maverling. Pittsburgh: Dorrance, 1994

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. Leonard, Robert J. “From Parlor Games to Social Science: Von Neumann, Morgenstern and the Creation of Game Theory, 1928–1944.” Journal of Economic Literature (1995). ———. “Reading Cournot, Reading Nash: The Creation and Stabilization of the Nash Equilibrium.” The Economic Journal (May 1994),

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gave much-appreciated advice and encouragement at every stage. Avinash Dixit, Harold Kuhn, Roger Myerson, Ariel Rubinstein, and Robert Wilson patiently shared their insights about game theory and served as valuable sounding boards. Donald Spencer, Harold Kuhn, Lars Hormander, Michael Artin, Joseph Kohn, John Milnor, Louis Nirenberg, and Jürgen Moser worked

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, 221 Econometrica, 91, 120 Econometric Society, 20 Nash’s fellowship in, 354–55 economics: bargaining and, 88–91, 120, 129, 149–151, 360 see also game theory; Nobel Prize in economics Edgeworth, Francis Ysidro, 88, 89 Ehrlich, Phillip, 287 Eilenberg, Samuel, 68 Einstein, Albert, 12, 13, 15, 19, 41, 46, 50,

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131 National Science Foundation (NSF), 107, 236, 296, 313, 314 Navier-Stokes equations, 297 Navy, U.S., 82, 83, 125, 126, 134, 135 negotiation, in game theory, 120 Nehru, Jawaharlal, 278 Nelson, Ed, 284, 286, 296, 300 Nerval, Gerard de, 228 Neuwirth, Jerome, 144, 182, 231 New Jersey Transit, 346 Newman, Donald

The Ghost Map: A Street, an Epidemic and the Hidden Power of Urban Networks.

by Steven Johnson  · 18 Oct 2006  · 304pp  · 88,773 words

prominent avenue that you know is only a few feet away. And indeed, the street layout was explicitly designed to serve as a barricade. When John Nash designed Regent Street to connect Marylebone Park with the Prince Regent’s new home at Carlton House, he planned the thoroughfare as a kind of

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-era nuclear politics. Mutually assured destruction isn’t much of a deterrent to him. Mutually assured destruction, in fact, sounds like a pretty good outcome. Game theory has always had trouble accounting for players with no rational self-interest, and the theories of nuclear deterrence are no exception. And once the bomb

The Mathematics of Love: Patterns, Proofs, and the Search for the Ultimate Equation

by Hannah Fry  · 3 Feb 2015  · 88pp  · 25,047 words

who has seen the 2001 film A Beautiful Mind might think that maths already has the answer. The film follows the life of mathematics superstar John Nash and includes some dramatized explanations of his major mathematical breakthroughs. In one famous scene, Nash and his three charming gentlemen friends spot a group of

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ends up better off if they ignore their personal preferences. The mathematics hidden behind this problem is game theory – a way to formalize strategies and find the best tactic in a situation. Despite the name, game theory isn’t just about studying activities undertaken for amusement. It can apply in any situation where two

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in a group, and generally it’s difficult to persuade people to ignore those preferences for the greater good. So maybe we have to leave game theory behind for the time being. But that doesn’t mean that maths doesn’t have any tips to help you on your night out. To

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are based on a single idea: how to exploit stereotypes to try and maximize your own reward. As we’ve already seen, the mathematics of game theory can be used to beat other suitors. And if you’re looking to turn the dating game into a dating war, it is also ideally

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placed to provide the best strategy in a romantic contest between two opponents. A warning: game theory encourages you to exploit the weaknesses of your opponents. When applied to dating, this view comes with a slightly cynical picture of the world. As

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a result, the first half of this chapter will show you some of the best tenets of game theory, not the best tenets of human morality. And because they rely on exploiting the supposed differences between men and women, they don’t really work

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dating conundrums, whatever kind of relationship you’re in, towards the end of the chapter. But first, let me begin with an illustration of how game theory can be used by men with only one thing on their mind. Getting what you want out of women Gentlemen: your challenge, if you choose

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. Your task is to choose which gifts to give that are most likely to get you your reward, but without attracting dangerous gold-diggers. Meanwhile, game theory has given you an opponent: the woman, who can decide whether or not to accept the gift. Her task is to try and capture the

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), but these assumptions do make for a wonderfully neat mathematical problem. The full derivation of the best strategy for the man7 gets quite heavily into game theory at points and is not for the fainthearted, but the result is a great example of the theory in action. And the best strategy for

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left out with this example being so focused on what the men should do. But fear not, there are plenty of slightly patronizing applications of game theory out there to help you snare your own rewards, too. Because if men are only after sex, of course we women are constantly trying to

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there be the same number of both? In what became known as the eligible bachelor paradox, Mark Gimein offered an answer to this question using game theory, with a set of assumptions as follows. Throughout the course of his life, a man will date a range of women. Because of their looks

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’s often the bidder in the weaker position who comes away with the prize, a phenomenon which has been the subject of extensive attention within game-theory literature. As with the previous example, the theory8 gets quite heavy in places, but the insights can go some way toward explaining why there are

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alone and rush out to buy a houseful of cats, it’s worth pausing and looking at these examples objectively. As neat an application of game theory as they are mathematically, they have one flawed assumption at their core: that men are trying to trick women into having sex with them and

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there may even be some women who want sex and some men who want commitment. And thus this particular game-theory house of cards comes tumbling down. Thankfully, there are ways to use game theory that don’t require men and women to conform to stereotypes, and in particular, a formulation that can apply

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betray each other. Using this set-up makes the game of being faithful equivalent to one of the most famous and well-studied problems in game theory: the prisoner’s dilemma. In the prisoner’s dilemma, two suspects are being questioned separately about the same crime. They have two choices: to cooperate

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cheating given how much you stand to lose in the long run. These ideas were first presented in Robert Axelrod’s groundbreaking 1984 book on game theory, The Evolution of Cooperation. In it, Axelrod explains how and why cooperation can occur in human and animal societies despite how stark things in Don

Explaining Humans: What Science Can Teach Us About Life, Love and Relationships

by Camilla Pang  · 12 Mar 2020  · 256pp  · 67,563 words

bonds, fundamental forces and human connection 10. How to learn from your mistakes Deep learning, feedback loops and human memory 11. How to be polite Game theory, complex systems and etiquette Afterword Acknowledgements Index About the Author Dr Camilla Pang holds a PhD in Biochemistry from University College London and is a

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human relationships and interaction; and machine learning can help us to make more organized decisions. Thermodynamics explains the struggle to create order in our lives; game theory provides a path through the maze of social etiquette; and evolution demonstrates why we have such strong differences in opinion. By understanding scientific principles, we

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the flow and temperature of water, and the volume of detergent, during the cycle according to how dirty clothes are. It also has applications in game theory and conflict resolution, as a methodology for mapping an ecosystem of different people with varying preferences, which may fluctuate between 0 and 1 – from absolute

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the adjustments over time that allow us to get the most out of this powerful, sometimes dangerous, source of ourselves. 11. How to be polite Game theory, complex systems and etiquette ‘Hi, Millie, I’m calling to see if your mum is there?’ ‘Yes, she is,’ I said, and then put the

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me what the laws of etiquette were, I would have to work them out for myself. In doing so, relying on techniques from computer modelling, game theory and my own field of bioinformatics, I have learned that a rulebook is perhaps the wrong way to think about etiquette. Because the rules are

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others around them. It’s between these individual needs, local connections and global norms that the actual etiquette of a system is to be found. Game theory ABM can help you to discover what the etiquette is in a particular context. But it doesn’t tell you anything about why people behave

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a system interact, but what their motivations are, and why they make certain decisions. Game theory was pioneered by two mathematicians whose work helped lay the foundations for the modern study of artificial intelligence: John von Neumann and John Nash. Like agent-based models, it looks at how different players within a certain, rules

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-based system interact. But it goes further by looking at the consequences of their various choices: how will a decision by one or several players in the game affect everyone else? Game theory looks at

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and their consequences, but those of the other players as well – predicting both what they may know, and how they are likely to act. Among game theory’s many ideas and applications is the Nash equilibrium. This is the concept that, in any finite game, there is a point of balance where

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with it encourages us to look beyond our own perception of certain events, and to put ourselves in the shoes of the other player. Because game theory is ultimately about interdependence – how our outcome depends in part on someone else’s choices – we can’t just live in our own heads, or

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, in which everyone gets what they want without having to change course? If ABM allows you to understand the implied etiquette of a given system, game theory is the technique to model your subsequent decisions, aligning them with both your own ideal outcomes, and the choices others are making in parallel or

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choosing the direction either of Nash equilibrium (mutual benefit) or, if you want to be non-cooperative, individual advancement. I’ve come to rely on game theory for explaining why certain behaviour exists and overcoming my inability to detect a person’s motives (especially as these are rarely made explicit). It might

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phrase – ‘Having a good hair day, I see!’ – fell flat. I had failed to take into account that he was bald, shiningly so. For me, game theory is less about winning and more about surviving the life experiences that nothing has prepared me for. I don’t want to beat the other

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board without sending too many of them flying, like my mum’s poor friend at the Christmas party. This is the counter-intuitive benefit of game theory. While ostensibly being a playbook for rational decision making, it also reminds us of its limits. If we put everything in our lives through the

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lens of game theory, then we would end up in something like the dystopia that Thomas Hobbes outlined in Leviathan, as the fate of humanity without a body politic

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, poore, nasty, brutish and short’: this was the ‘state of nature’ he believed could only be counteracted by the creation of the centralized state. As game-theory junkies, we would become pure Homo economicus: totally self-interested players animated only by our search for what Hobbes characterized as felicity – the pursuit of

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desires for power and self-advancement (not, as I thought when I first read it, anything to do with the Austin Powers character Felicity Shagwell). Game theory could easily become the mechanism to fulfil Hobbes’s negative assessment of humans – as creatures that have to be prevented from harming both themselves and

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reciprocans, a person who wants to cooperate with others in pursuit of mutual benefit. The existence of Nash equilibria shows that the ultimate lesson of game theory is interdependence: we are all on the same board, playing the same game, and often depend on the help and support of others to achieve

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our desired outcome. Game theory could be a selfishness charter, but it’s also one of the best frameworks I know for demonstrating how we are all part of the

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carer. It’s these minor gestures – the ones that don’t immediately benefit us – that make us a social species and not an individualistic one. Game theory, which doesn’t have to be about competition, is one of the most important techniques for finding the common ground that defines our relationships as

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most efficient route to goal. And that is why etiquette really matters. Homology If agent-based modelling can help us to understand local context, and game theory to plot our own paths alongside those of others, the third leg to my etiquette stool is homology: the science of modelling connections and similarities

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me with the safest ground to tread on. Once actually in that environment, I can start to study the agents and model it accordingly, bringing game theory into play to decide exactly what to say and how to act. But it is homology that allows me to take the crucial first steps

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, 69 error, learning to embrace 21–4 fear and 71, 81 feedback loop and 199, 202, 203–204 fuzzy logic and 146, 156–60, 158 game theory and 215–18, 220 goals and 128–30, 134, 137, 138–41, 142, 143, 191 gradient descent algorithm and 138–41, 143, 191 homology and

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empathy and 149 ergodicity and 118, 120 evolution and xii, 31, 45–7, 118, 120, 146, 147, 148, 148, 149 fuzzy logic and 157, 162 game theory and 219 harmony and 104–105 hierarchy and 36 homology and 221–2 human survival and 118, 120 order and 61 probability and 154, 155

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40–41 entropy 48–9, 54–6, 57–8, 90 equilibrium achieving 64–7 Bayes’ theorem and 155 feedback and 202 fuzzy logic and 156 game theory and/Nash equilibrium 215, 216, 217 harmonic motion and 89, 90, 90–91 interference and 94, 95, 96 perfection and 50 resonance 97 ergodic theory

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full stop 107–108 fundamental forces, the four 174–80 fuzzy logic 146, 156–60, 158, 162 GAD (generalized anxiety disorder) x–xi, 74, 197 game theory xii, 157, 209, 215–19, 222 gamma waves 98 gene sequences 31 Gibbs free energy 55–6, 65 goals, achieving 122–43 anxiety, positive results

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170, 171, 172, 174, 175, 176, 181 politeness/etiquette 206–23 agent-based modelling (ABM) and 210–14, 215, 216 dating and 207, 221–2 game theory and 209, 215–19 homology and 219–22 pollen 112, 113 position thinking 129–30, 129, 131 probability 137, 146, 159, 161, 162, 206 empathy

Networks, Crowds, and Markets: Reasoning About a Highly Connected World

by David Easley and Jon Kleinberg  · 15 Nov 2010  · 1,535pp  · 337,071 words

5.4 A Weaker Form of Structural Balance . . . . . . . . . . . . . . . . . . . . . . 140 5.5 Advanced Material: Generalizing the Definition of Structural Balance . . . . 143 5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 II Game Theory 163 6 Games 165 6.1 What is a Game? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 6.2 Reasoning about Behavior in a Game . . . . . . . . . . . . . . . . . . . . . . 168 6.3 Best Responses and

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. . . . . . . . . . . . . . 189 6.9 Pareto-Optimality and Social Optimality . . . . . . . . . . . . . . . . . . . . 194 6.10 Advanced Material: Dominated Strategies and Dynamic Games . . . . . . . 196 6.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7 Evolutionary Game Theory 219 7.1 Fitness as a Result of Interaction . . . . . . . . . . . . . . . . . . . . . . . . 220 7.2 Evolutionarily Stable Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.3 A General Description of Evolutionarily Stable Strategies

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7.4 Relationship Between Evolutionary and Nash Equilibria . . . . . . . . . . . . 228 7.5 Evolutionarily Stable Mixed Strategies . . . . . . . . . . . . . . . . . . . . . 230 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 8 Modeling Network Traffic using Game Theory 239 8.1 Traffic at Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 8.2 Braess’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.3 Advanced Material: The Social Cost of Traffic at Equilibrium . . . . . . . .

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be used to reason about how fissures in a network may arise from the dynamics of conflict and antagonism at a purely local level. Game Theory. Our discussion of game theory starts from the observation that there are numerous settings in which a group of people must simultaneously choose how to act, knowing that

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e/e1/Late Medieval Trade Routes.jpg.) directly generalized to more complex patterns of interactions on networks. As a general part of our investigation of game theory, we will abstract such situations with inter-dependent behavior into a common framework, where a collection of individuals must each commit to a strategy,

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change his or her strategy, even knowing how others will behave. Markets and Strategic Interaction on Networks. Once we’ve developed graph theory and game theory, we can combine them to produce richer models of behavior on networks. One natural setting where we can explore this is in models of trade

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explanation for your answer. (Hint: Think about any unbalanced triangle in the network, and how X must attach to the nodes in it.) Part II Game Theory 163 Chapter 6 Games In the opening chapter of the book, we emphasized that the “connectedness” of a complex social, natural, or technological system

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theory. In this second part of the book, we study interconnectedness at the level of behavior, developing basic models for this in the language of game theory. Game theory is designed to address situations in which the outcome of a person’s decision depends not just on how they choose among several options, but

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where adoption decisions are affected by what others are doing. As a first step, then, we begin with a discussion of the basic ideas behind game theory. For now, this will involve descriptions of situations in which people interact with one another, initially without an accompanying graph structure. Once these ideas

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into the picture in subsequent chapters, and begin to consider how structure and behavior can be studied simultaneously. 6.1 What is a Game? Game theory is concerned with situations in which decision-makers interact with one another, and in which the happiness of each participant with the outcome depends not

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in Section 6.1, this means that the two players are solely concerned with maximizing their own average grade. However, nothing in the framework of game theory requires that players care only about personal rewards. For example, a player who is altruistic may care about both his or her own benefits,

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need the full technical content of common knowledge in anything we do here, it is an underlying assumption and a topic of research in the game theory literature [28]. As mentioned earlier, it is still possible to analyze games in situations where common knowledge does not hold, but the analysis becomes

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is a strict best response to C. So how should we reason about the outcome of play in this game? Defining Nash Equilibrium. In 1950, John Nash proposed a simple but powerful principle for reasoning about behavior in general games [307, 308], and its underlying premise is the following: even when there

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by enlarging the set of strategies to include the possibility of randomization; once players 184 CHAPTER 6. GAMES are allowed to behave randomly, one of John Nash’s main results establishes that equilibria always exist [307, 308]. Probably the simplest class of games to expose this phenomenon are what might be called

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penalty kicks in soccer as a two-player game. In 2002, Ignacio Palacios-Huerta undertook a large study of penalty kicks from the perspective of game theory [331], and we focus on his analysis here. As he observed, penalty kicks capture the ingredients of two-player, two-strategy games remarkably faithfully.

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it cooperates. Given this, Firm 1 can predict that it is safe to 6.10. ADVANCED MATERIAL: DOMINATED STRATEGIES AND DYNAMIC GAMES209 enter. In game theory, the standard model for dynamic games in extensive form assumes that players will seek to maximize their payoff at any intermediate stage of play that

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reason about what the other players may do. In this chapter, on the other hand, we explore the notion of evolutionary game theory, which shows that the basic ideas of game theory can be applied even to situations in which no individual is overtly reasoning, or even making explicit decisions. Rather, game-theoretic

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the population. In this way, fitter genes tend to win over time, because they provide higher rates of reproduction. The key insight of evolutionary game theory is that many behaviors involve the interaction of multiple organisms in a population, and the success of any one of these organisms depends on how

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: Reasoning about a Highly Connected World. To be published by Cambridge University Press, 2010. Draft version: September 29, 2009. 219 220 CHAPTER 7. EVOLUTIONARY GAME THEORY an organism’s genetically-determined characteristics and behaviors are like its strategy in a game, its fitness is like its payoff, and this payoff depends

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play one of these two strategies through its whole lifetime. As a result, the idea of choosing strategies — which was central to our formulation of game theory — is missing from the biological side of the analogy. Instead of a notion of choice among strategies, evolutionary ideas will supply a different ingredient

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that each beetles is repeatedly paired off with other beetles in food competitions over the course of its lifetime. We will 222 CHAPTER 7. EVOLUTIONARY GAME THEORY assume the population is large enough that no two particular beetles have a significant probability of interacting with each other repeatedly. A beetle’s overall

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. As a result, the population of large beetles resists the invasion of small beetles, and so Large is evolutionarily stable. 224 CHAPTER 7. EVOLUTIONARY GAME THEORY Therefore, if we know that the large-body-size mutation is possible, we should expect to see populations of large beetles in the wild, rather

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in biology, it can be applied in many other contexts. For example, suppose a large group of people are being 230 CHAPTER 7. EVOLUTIONARY GAME THEORY matched repeatedly over time to play the General Symmetric Game from Figure 7.3. Now the payoffs should be interpreted as reflecting the welfare of

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evolutionary forces, we need to generalize the notion of evolutionary stability by allowing some notion of “mixing” between strategies. Defining Mixed Strategies in Evolutionary Game Theory. There are at least two natural ways to introduce the idea of mixing into the evolutionary framework. First, it could be that each individual is

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which organisms must break the symmetry between two distinct behaviors, when consistently adopting just one of these behaviors is evolutionarily unsustainable. 234 CHAPTER 7. EVOLUTIONARY GAME THEORY We can interpret the result of this example in two possible ways. First, all participants in the population may actually be mixing over the two

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Hawk-Dove, and what the evolutionary consequences might be for the observed behavior within a lion population. In this, as in many examples from evolutionary game theory, it is beyond the power of current empirical studies to work out detailed fitness values for particular strategies. However, even in situations where exact

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(You do not need to write a formal proof; a careful explanation is fine.) 238 CHAPTER 7. EVOLUTIONARY GAME THEORY Chapter 8 Modeling Network Traffic using Game Theory Among the initial examples in our discussion of game theory in Chapter 6, we noted that traveling through a transportation network, or sending packets through the Internet, involves

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to proving the easier but weaker factor of 2 between the social optimum and some equilibrium traffic pattern. 246 CHAPTER 8. MODELING NETWORK TRAFFIC USING GAME THEORY players’ strategies by constantly having some player perform his or her best response to the current situation. If the procedure ever stops, in a

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a collective activity (such as network exchange in the present case). In this framework, stability can be formulated using a central notion in cooperative game theory known as the core solution, and balance can be formulated as a combination of the core solution and a second notion known as the kernel

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shared resource. We argued that the Nash bargaining solution provides a natural prediction for how the surplus available in the bargaining will be divided. When John Nash originally formulated this notion, he motivated it by first writing down a set of axioms he believed the outcome of any bargaining solution should satisfy

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references, pursuing serendipitious leads from one topic to another. For example, Figure 13.4 shows the cross-references among Wikipedia articles on certain topics in game theory, together with connections to related topics.1 We can see, for example, 1Since Wikipedia changes constantly, Figure 13.4 necessarily represents the state of

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information network that can be represented as a directed graph. The figure shows the cross-references among a set of Wikipedia articles on topic in game theory, and their connections to related topics including popular culture and government agencies. how it’s possible to get from the article on Nash Equilibrium

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to the article on NASA (the U.S. National Aeronautics and Space Administration) by passing through articles on John Nash (the creator of Nash equilibrium), A Beautiful Mind (a film about John Nash’s life), Ron Howard (the director of A Beautiful Mind), Apollo 13 (another film directed by Ron Howard), and

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only short chain of articles from Nash equilibrium to NASA. Figure 13.4 also contains a sequence of cross-references based on the fact that John Nash worked for a period of time at RAND, and RAND is the subject of several conspiracy theories, as is NASA. These short paths between

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the Voting Experiment. Once we appreciate what’s going on here, it becomes natural to think of voting with a shared objective in terms of game theory. The voters correspond to players, their possible strategies are the possible ways of choosing votes based on private information, and they receive payoffs based

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[91] Robert B. Cairns and Beverly D. Cairns. Lifelines and Risks: Pathways of Youth in our Time. Cambridge University Press, 1995. [92] Colin Camerer. Behavioral Game Theory: Experiments in Strategic Interaction. Princeton University Press, 2003. [93] Rebecca L. Cann, Mark Stoneking, and Allan C. Wilson. Mitochondrial DNA and human evolution. Nature, 325

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1998. [170] Eric Friedman, Paul Resnick, and Rahul Sami. Manipulation-resistant reputation systems. In Noam Nisan, Tim Roughgarden, Éva Tardos, and Vijay Vazirani, editors, Algorithmic Game Theory, pages 677–698. Cambridge University Press, 2007. [171] Milton Friedman. Essays in Positive Economics. University of Chicago Press, 1953. [172] H. L. Frisch and J

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, 84(2):427–436, September 1978. 816 BIBLIOGRAPHY [230] Yakar Kannai. The core and balancedness. In Robert J. Aumman and Sergiu Hart, editors, Handbook of Game Theory, volume 1, pages 355–395. Elsevier, 1992. [231] Michael L. Katz and Carl Shapiro. Network externalities, competition, and compatibility. American Economic Review, 75(3

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Microeconomic Theory. Oxford University Press, 1995. [284] Michael Maschler. The bargaining set, kernel, and nucleolus. In Robert J. Aumman and Sergiu Hart, editors, Handbook of Game Theory, volume 1, pages 592–667. Elsevier, 1992. [285] Doug McAdam. Recruitment to high-risk activism: The case of Freedom Summer. American Journal of Sociology, 92

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Myerson. Incentive compatibility and the bargaining problem. Econometrica, 47:61–73, 1979. [306] John Nash. The bargaining problem. Econometrica, 18:155–162, 1950. [307] John Nash. Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA, 36:48–49, 1950. [308] John Nash. Non-cooperative games. Annals of Mathematics, 54:286–295, 1951. [309] National

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and Chris Canning, editors, Handbook of Statistical Genetics, pages 179–212. John Wiley & Sons, 2001. [320] Martin A. Nowak and Karl Sigmund. Phage-lift for game theory. Nature, 398:367–368, April 1999. [321] Martin A. Nowak and Karl Sigmund. Evolutionary dynamics of biological games. Science, 303:793–799, February 2004. [322

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patterns and business models for the next generation of software. Communication and Strategy, 1:17, 2007. [330] Martin Osboren and Ariel Rubinstein. A Course in Game Theory. The MIT Press, 1994. [331] I. Palacios-Huerta. Professionals play minimax. Review of Economic Studies, 70:395–415, 2003. BIBLIOGRAPHY 823 [332] Christopher R.

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Law Review, 44(309), 2007. [372] Tamás Solymosi and Tirukkannamangai E. S. Raghavan. An algorithm for finding the nucleolus of assignment games. International Journal of Game Theory, 23:119–143, 1994. [373] Michael Spence. Job market signaling. Quarterly Journal of Economics, 87:355–374, 1973. [374] Olaf Sporns, Dante R. Chialvo,

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] Éva Tardos and Tom Wexler. Network formation games and the potential function method. In Noam Nisan, Tim Roughgarden, Éva Tardos, and Vijay Vazirani, editors, Algorithmic Game Theory, pages 487–516. Cambridge University Press, 2007. [380] Richard H. Thaler. Anomalies: The ultimatum game. Journal of Economic Perspectives, 2(4):195–206, 1988. [

Artificial Intelligence: A Modern Approach

by Stuart Russell and Peter Norvig  · 14 Jul 2019  · 2,466pp  · 668,761 words

.4Partially Observable MDPs 16.5Algorithms for Solving POMDPs Summary Bibliographical and Historical Notes 17Multiagent Decision Making 17.1Properties of Multiagent Environments 17.2Non-Cooperative Game Theory 17.3Cooperative Game Theory 17.4Making Collective Decisions Summary Bibliographical and Historical Notes 18Probabilistic Programming 18.1Relational Probability Models 18.2Open-Universe Probability Models 18.3Keeping Track of

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a game: the actions of one player can significantly affect the utility of another (either positively or negatively). Von Neumann and Morgenstern’s development of game theory (see also Luce and Raiffa, 1957) included the surprising result that, for some games, a rational agent should adopt policies that are (or least appear

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to be) randomized. Unlike decision theory, game theory does not offer an unambiguous prescription for selecting actions. In AI, decisions involving multiple agents are studied under the heading of multiagent systems (Chapter 17

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but include a component for the size of the belief state—smaller is better! We will return to partially observable games under the topic of Game Theory in Section 17.2. 6.6.2Card games Card games such as bridge, whist, hearts, and poker feature stochastic partial observability, where the missing information

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16 describes techniques for stochastic environments (in which outcomes of actions have probabilities associated with them): Markov decision processes, partially observable Markov decision processes, and game theory. In Chapter 23 we show that reinforcement learning allows an agent to learn how to behave from past successes and failures. Bibliographical and Historical Notes

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dynamic programming algorithm gives an efficient solution for both evidence subset election and conditional plan generation. Harsanyi (1967) studied the problem of incomplete information in game theory, where players may not know each others’ payoff functions exactly. He showed that such games were identical to games with imperfect information, where players are

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fact that these other agents are also taking into account the preferences of other agents, and so on. This brings us into the realm of game theory: the theory of strategic decision making. It is this strategic aspect of reasoning—players each taking into account how other players may act—that distinguishes

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to analyze its possible decisions and compute the expected utility for each of these (under the assumption that other agents are acting rationally, according to game theory). In this way, game-theoretic techniques can determine the best strategy against a rational player and the expected return for each player. 2.Mechanism design

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so that the collective good of all agents is maximized when each agent adopts the game-theoretic solution that maximizes its own utility. For example, game theory can help design the protocols for a collection of Internet traffic routers so that each router has an incentive to act in such a way

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that global throughput is maximized. Mechanism design can also be used to construct intelligent multiagent systems that solve complex problems in a distributed fashion. Game theory provides a range of different models, each with its own set of underlying assumptions; it is important to choose the right model for each setting

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In the world of computer programs, it may be possible to inspect source code to make sure it will follow an agreement. We use cooperative game theory to analyze this situation. •If binding agreements are not possible, we have a non-cooperative game. Although this term suggests that the game is inherently

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guarantees cooperation. But it could well be that agents independently decide to cooperate, because it is in their own best interests. We use non-cooperative game theory to analyze this situation. Some environments will combine multiple different dimensions. For example, a package delivery company may do centralized, offline planning for the routes

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value for each player—if players use mixed strategies, then we must use expected utility. So, how should agents decide act in games like Morra? Game theory provides a range of solution concepts that attempt to define rational action with respect to an agent’s beliefs about the other agent’s beliefs

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part of the strategy profile that determines the outcome. We introduce our first solution concept through what is probably the most famous game in the game theory canon—the prisoner’s dilemma. This game is motivated by the following story: Two alleged burglars, Ali and Bo, are caught red-handed near

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at least one strategy profile and no worse on any other. A dominant strategy is a strategy that dominates all others. A common assumption in game theory is that a rational player will always choose a dominant strategy and avoid a dominated strategy. Being rational—or at least not wishing to be

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see that Bo playing heads with probability 0.6 could not form part of any Nash equilibrium. 17.2.2Social welfare The main perspective in game theory is that of players within the game, trying to obtain the best outcomes for themselves that they can. However, it is sometimes instructive to

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round, earning a total jail sentence of 500 years each. This type of reasoning is known as backward induction, and plays a fundamental role in game theory. However, if we drop one of the three conditions—fixed, finite, or mutually known—then the inductive argument doesn’t hold. Suppose that the

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of the hard cases from the list of environment properties on page 61. However, there are two limitations to the extensive form in particular and game theory in general. First, it does not deal well with continuous states and actions (although there have been some extensions to the continuous case; for

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of the game over time, the model begins to break down. Let’s examine each source of uncertainty, and whether each can be represented in game theory. Actions: There is no easy way to represent a game where the players have to discover what actions are available. Consider the game between computer

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virus writers and security experts. Part of the problem is anticipating what action the virus writers will try next. Strategies: Game theory is very good at representing the idea that the other players’ strategies are initially unknown—as long as we assume all agents are rational. The

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size of the tree every time we add another node; a habit that quickly leads to intractably large trees. Because of these and other problems, game theory has been used primarily to analyze environments that are at equilibrium, rather than to control agents within an environment. 17.2.5Uncertain payoffs and assistance

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we expect the key property of assistance games to remain true: the more intelligent the robot, the better the outcome for the human. 17.3Cooperative Game Theory Recall that cooperative games capture decision making scenarios in which agents can form binding agreements with one another to cooperate. They can then benefit from

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always arrive at a collectively desirable Pareto–optimal outcome in the prisoner’s dilemma. 17.3.2Strategy in cooperative games The basic assumption in cooperative game theory is that players will make strategic decisions about who they will cooperate with. Intuitively, players will not desire to work with unproductive players—they will

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is likely in practice. 17.4.4Bargaining Bargaining, or negotiation, is another mechanism that is used frequently in everyday life. It has been studied in game theory since the 1950s and more recently has become a task for automated agents. Bargaining is used when agents need to reach agreement on a matter

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in which no agent has an incentive to deviate from its specified strategy. We have techniques for dealing with repeated games and sequential games. •Cooperative game theory considers settings in which agents can make binding agreements to form coalitions in order to cooperate. Solution concepts in cooperative game attempt to formulate which

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in the late 1980s, when it was widely realized that agents with differing preferences are the norm in AI and society—from this point on, game theory began to be established as the main methodology for studying such agents. Multiagent planning has leaped in popularity in recent years, although it does have

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rewards for every collision. Each global decision maximizes the sum of Q-functions and the whole process converges to globally optimal solutions. The roots of game theory can be traced back to proposals made in the 17th century by Christiaan Huygens and Gottfried Leibniz to study competitive and cooperative human interactions scientifically

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and mathematically. Throughout the 19th century, several leading economists created simple mathematical examples to analyze particular examples of competitive situations. The first formal results in game theory are due to Zermelo (1913) (who had, the year before, suggested a form of minimax search for games, albeit an incorrect one). Emile Borel (1921

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the Theory of Games and Economic Behavior, the defining book for game theory. Publication of the book was delayed by the wartime paper shortage until a member of the Rockefeller family personally subsidized its publication. In 1950, at the age of 21, John Nash published his ideas concerning equilibria in general (non-zero-sum

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Harsanyi) in 1994. The Bayes–Nash equilibrium is described by Harsanyi (1967) and discussed by Kadane and Larkey (1982). Some issues in the use of game theory for agent control are covered by Binmore (1982). Aumann and Brandenburger (1995) show how different equilibria can be reached depending on the knowleedge each player

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server running for 8 days to compute a baseline strategy for their Pluribus program. With that strategy they were able to defeat human champion opponents. Game theory and MDPs are combined in the theory of Markov games, also called stochastic games (Littman, 1994; Hu and Wellman, 1998). Shapley (1953b) actually described the

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and Sudholter (2002). Simple games in general are discussed in detail by Taylor and Zwicker (1999). For an introduction to the computational aspects of cooperative game theory, see Chalkiadakis et al. (2011). Many compact representation schemes for cooperative games have been developed over the past three decades, starting with the work of

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same name. The ACM Conference on Electronic Commerce (EC) also publishes many relevant papers, particularly in the area of auction algorithms. The principal journal for game theory is Games and Economic Behavior. 1Morra is a recreational version of an inspection game. In such games, an inspector chooses a day to inspect a

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discriminator and the discriminator learning to accurately separate real from fake data. The competition between generator and discriminator can be described in the language of game theory (see Chapter 17). The idea is that in the equilibrium state of the game, the generator should reproduce the training distribution perfectly, such that the

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safely. Robotics brings together many of the concepts we have seen in this book, including probabilistic state estimation, perception, planning, unsupervised learning, reinforcement learning, and game theory. For some of these concepts robotics serves as a challenging example application. For other concepts this chapter breaks new ground, for instance in introducing the

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can be handled by MDPs), partial observability (which can be handled by POMDPs), and acting with and around other agents (which can be handled with game theory). The problem is made even harder by the fact that most robots work in continuous and high-dimensional state and action spaces. They also operate

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when there are multiple factors to consider. In principle, utility-maximization agents address those issues in a completely general way. The fields of economics and game theory, as well as AI, make use of this insight: just declare what you want to optimize, and what each action does, and we can

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., Szerlip, P, Horsfall, P., and Goodman, N. D. (2019). Pyro: Deep universal probabilistic programming. JMLR, 20, 1–26. Binmore, K. (1982). Essays on Foundations of Game Theory. Pitman. Biran, O. and Cotton, C. (2017). Explanation and justification in machine learning: A survey. In Proc.IJCAI-17 Workshop on Explainable AI. Bishop, C

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Sarkar, S. C. (1989). Heuristic search in restricted memory. AIJ, 41, 197–222. Chalkiadakis, G., Elkind, E., and Wooldridge, M. (2011). Computational Aspects of Cooperative Game Theory. Morgan Kaufmann. Chalmers, D. J. (1992). Subsymbolic computation and the Chinese room. In Dinsmore, J. (Ed.), The symbolic and connectionist paradigms: Closing the gap. Lawrence

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planning system for generating spacecraft mission plans. In First International Conference on Expert Planning Systems. Institute of Electrical Engineers. Fudenberg, D. and Tirole, J. (1991). Game theory. MIT Press. Fukunaga, A. S., Rabideau, G., Chien, S., and Yan, D. (1997). ASPEN: A framework for automated planning and scheduling of spacecraft control and

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regularities in sparse and explicit word representations. In Proc. Eighteenth Conference on Computational Natural Language Learning. Leyton–Brown, K. and Shoham, Y. (2008). Essentials of Game Theory: A Concise, Multidisciplinary Introduction. Morgan & Claypool. Li, C. M. and Anbulagan (1997). Heuristics based on unit propagation for satisfiability problems. In IJCAI-97. Li, K

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. (1981). Optimal auction design. Mathematics of Operations Research, 6, 58–73. Myerson, R. (1986). Multistage games with communication. Econometrica, 54, 323–358. Myerson, R. (1991). Game Theory: Analysis of Conflict. Harvard University Press. Nair, V. and Hinton, G. E. (2010). Rectified linear units improve restricted Boltzmann machines. In ICML-10. Nalwa, V

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effectively. Distill, 1. Waugh, K., Schnizlein, D., Bowling, M., and Szafron, D. (2009). Abstraction pathologies in extensive games. In AAMAS–09. Weibull, J. (1995). Evolutionary Game Theory. MIT Press. Weidenbach, C. (2001). SPASS: Combining superposition, sorts and splitting. In Robinson, A. and Voronkov, A. (Eds.), Handbook of Automated Reasoning. MIT Press. Weiss

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desires and, 518–519 propagation, 476 loopy, 476 revision, 353 update, 353 belief network, see Bayesian network belief state, 140, 259, 383, 403, 406 in game theory, 610 probabilistic, 479, 483 wiggly, 261 Belkin, M., 734, 1087 Bell, C., 377, 401, 1087 Bell, D. A., 798, 1090 Bell, J. L., 297,

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, 224 Scrabble, 225 Settlers of Catan, 197 stochastic, 210 Tetris, 562, 571 Yahtzee, 214 zero-sum, 193, 600 game playing, 192–221 game show, 524 game theory, 28, 590, 635 cooperative, 616 non-cooperative, 595–615 Gammage, C., 48, 1113 GAN (generative adversarial network), 831, 838, 1022 Ganchev, K., 904, 1085 Gandomi

The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling

by Adam Kucharski  · 23 Feb 2016  · 360pp  · 85,321 words

replace, we are less likely to insure it. Over the following chapters, we will find out how gambling has continued to influence scientific thinking, from game theory and statistics to chaos theory and artificial intelligence. Perhaps it shouldn’t be surprising that science and gambling are so intertwined. After all, wagers are

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cigarette advertising had fallen by over 25 percent. Yet tobacco revenues held steady. Thanks to the government, the equilibrium had been broken. JOHN NASH PUBLISHED HIS first papers on game theory while he was a PhD student at Princeton. He’d arrived at the university in 1948, after being awarded a scholarship on the

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to von Neumann and Morgenstern, there was now mathematical proof that it was. Despite his fondness for Berlin’s nightlife, von Neumann didn’t use game theory when he visited casinos. He saw poker mainly as an intellectual challenge and eventually moved on to other problems. It would be several decades before

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unlike many competitors, Ferguson’s extraordinary success did not rely solely on intuition or instinct. When he played in the World Series, he was using game theory. The year before he beat Cloutier, Ferguson had completed a doctorate in computer science at UCLA. During that time, he worked as a consultant for

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as laying the foundations for the entire field, the answer would go on to cause a bitter dispute about who was the true inventor of game theory. GAMES LIKE POKER ARE “zero-sum,” with winning players’ profits equal to other players’ losses. When two players are involved, this means one person is

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optimal strategies always existed was a crucial breakthrough. He later said that without the result, there would have been no point continuing his work on game theory. The method von Neumann used to attack the minimax problem was far from simple. Lengthy and elaborate, it has been described as a mathematical “tour

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behind von Neumann’s minimax work had already been in place (though von Neumann had apparently been unaware of it). By applying the techniques to game theory, he said that von Neumann had “simply entered an open door.” The approaches Fréchet was referring to were the brainchild of his colleague Émile Borel

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von Neumann. When Borel’s papers were eventually published in English in the early 1950s, Fréchet wrote an introduction crediting him with the invention of game theory. Von Neumann was furious, and the pair exchanged barbed comments in the economics journal Econometrica. The dispute raised two important issues about applying mathematics to

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value become apparent. As Ferguson discovered when he applied game theory to poker, sometimes an idea that seems unremarkable to scientists can prove extremely powerful when used in a different context. While the fiery debate between von Neumann and Fréchet sparked and crackled, John Nash was busy finishing his doctorate at Princeton. By establishing

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players. The next problem is working out how to find them. MOST PEOPLE WHO HAVE a go at creating poker bots don’t rummage through game theory to find optimal strategies. Instead, they often start off with rule-based approaches. For each situation that could crop up in a game, the creator

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rates online, most creators aren’t very good at poker. BECAUSE RULE-BASED TACTICS CAN be difficult to get right, some people have turned to game theory to improve their computer players. But it’s tricky to find the optimal tactics for a game as complicated as Texas hold’em poker. Because

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against a perfect poker bot, a near-equilibrium strategy will struggle. But how easy is it to make a perfect bot for a complex game? GAME THEORY WORKS BEST in straightforward games in which all information is known. Tic-tac-toe is a good example: after a few games, most people work

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somewhere along the way. Unlike many game-playing programs, Chinook would often pick these antihuman strategies over an option that was actually better according to game theory. Chinook played its first tournament in 1990, coming in second in the US National Checker Championship. This should have meant it qualified for the World

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to a completely incompetent opponent. To understand why, we must look at another piece of Émile Borel’s research. As well as his work on game theory, Borel was interested in very unlikely events. To illustrate how seemingly rare things will almost certainly happen if we wait long enough, he coined the

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of moves but doesn’t show what those moves are. For instance, although strong solutions have been found for Connect Four and tic-tac-toe, John Nash showed in 1949 that when any such get-so-many-in-a-row-style game is played perfectly, the player who goes second will never

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is even greater in games like chess and poker, where nobody knows the perfect strategy. Which raises an important question: What happens when we apply game theory to games that are too complicated to fully learn? ALONG WITH TOBIAS GALLA, a physicist at the University of Manchester, Doyne Farmer has started to

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question how game theory holds up when games aren’t simple. Game theory relies on the assumption that all players are rational. In other words, they are aware of the effects of the various decisions

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wrong,” they said, “explaining why we are wrong will hopefully stimulate game theorists to think more carefully about the generic properties of real games.” ALTHOUGH GAME THEORY CAN help us identify the optimal strategy, it’s not always the best approach to use when players are error-prone or have to learn

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ensured the program picked strategies that would entice its opponents into making mistakes. Chris Ferguson was also aware of the issue. As well as employing game theory, he looked for changes in body language, adjusting his betting if players become nervous or overconfident. Players don’t just need to anticipate how the

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work has been years in the making. In 2003, an expert human player competed against one of the leading poker bots. Although the bot used game theory strategies to make decisions, it could not predict the changing behavior of its competitors. Afterward, the human player told the bot’s creators, “You have

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one of their games, Metropolis won ten dollars from von Neumann. He was delighted to beat a man who’d written an entire book on game theory. Metropolis used half the money to buy a copy of von Neumann’s Theory of Games and Economic Behavior and stuck the remaining five dollars

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inside the cover to mark the win. Even before von Neumann had published his book on game theory, his research into poker was well known. In 1937, von Neumann had presented his work in a lecture at Princeton University. Among the attendees, there

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, most people find that the computer is pretty hard to beat; play lots of games, and the computer will generally end up in the lead. Game theory suggests that if you follow the optimal strategy for rock-paper-scissors, and choose randomly between the three available options, you should expect to come

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brain.” When it comes to rock-paper-scissors, machines are much better than humans at coming up with the unpredictable moves required for an optimal game theory strategy. Such a strategy is inherently defensive, of course, because it aims to limit potential losses against a perfect opponent. But the rock-paper-scissors

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a picture of its opponent. Such approaches can be particularly important in games like poker, which can have more than two players. Recall that, in game theory, optimal strategies are said to be in Nash equilibrium: no single player will gain anything by picking a different strategy. Neil Burch, one of the

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tactics together. For instance, two of the players could decide to gang up on the third. When von Neumann and Morgenstern wrote their book on game theory, they noted that such coalitions work only when there are at least three players. “In a two-person game there are not enough players to

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, we would say that they cheat by co-operating, but if it happens just by accident, we would not.” That’s the problem with using game theory in poker: coalitions don’t always have to be deliberate. They might just result from the strategies players choose. In many situations, there is more

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selected tactics that just so happen to pick on the third player. This is why three-player poker is so difficult to tackle from a game theory point of view. Not only is the game far more complicated, with more potential moves to analyze, it’s not clear that hunting for the

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is always the best approach. “Even if you could compute one,” Michael Johanson said, “it wouldn’t necessarily be useful.” There are other drawbacks, too. Game theory can show you how to minimize your losses against a perfect opponent. But if your opponent has flaws—or if there are more than two

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the pinnacle of randomness, roulette was first beaten with statistics, and then with physics. Other games have fallen to science too. Poker players have explained game theory and syndicates have turned sports betting into investments. According to Stanislaw Ulam, who worked on the hydrogen bomb at Los Alamos, the presence of skill

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is helpful but by no means a sure route to victory. Gamblers also need to account for their opponents’ behavior. When John von Neumann developed game theory to tackle this problem, he found that employing deceptive tactics such as bluffing was actually the optimal thing to do. The gamblers had been right

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why. Sometimes it’s necessary to stray from mathematical perfection altogether. As researchers delve further into the science of poker, they are finding situations where game theory comes up short and where traditional gambling traits—reading opponents, exploiting weaknesses, spotting emotion—can help computer players become the best in the world. It

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led to the Monte Carlo method, now used in everything from 3D computer graphics to the analysis of disease outbreaks. And we have seen how game theory emerged from John von Neumann’s analysis of poker. The relationship between science and betting continues to thrive today. As ever, the ideas are flowing

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Marihuana and Drug Abuse, 1972). http://www.druglibrary.org/schaffer/library/studies/nc/nc2b.htm. 136Far from hurting tobacco companies’ profits: McAdams, David. Game-Changer: Game Theory and the Art of Transforming Strategic Situations (New York: W. W. Norton, 2014), 61. 137Yet tobacco revenues held steady: Hamilton, James. “The Demand for Cigarettes

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.” Review of Economics and Statistics 54, no. 4 (1972). 137“Mr. Nash is nineteen years old”: The letter was posted online by Princeton University after John Nash’s death in 2015. It went viral. 138Despite his prodigious academic record: Halmos, Paul. “The Legend of John von Neumann.” American Mathematical Monthly 8 (1973

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): 382–394. 138“Real life consists of bluffing”: Harford, Tim. “A Beautiful Theory.” Forbes, December 14, 2006. http://www.forbes.com/2006/12/10/business-game-theory-tech-cx_th_games06_1212harford.html. Original quote made in BBC show “Ascent of Man,” broadcast in 1973. 138Von Neumann started by looking at poker

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: Ferguson, Chris, and Thomas S. Ferguson. “On the Borel and von Neumann Poker Models.” Game Theory and Applications 9 (2003): 17–32. 139in a book titled Theory of Games and Economic Behavior: Von Neumann, John, and Oskar Morgenstern. Theory of Games

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): 1518–1522. doi:10.1126/science.1144079. 158John Nash showed in 1949: Demaine, Erik D., and Robert A. Hearn. “Playing Games with Algorithms: Algorithmic Combinatorial Game Theory.” Mathematical Foundations of Computer Science (2001): 18–32. http://erikdemaine.org/papers/AlgGameTheory_GONC3/paper.pdf. 159Twenty-six moves later: Schaeffer, Jonathan, and Robert Lake

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details in: Sandholm, T. “Perspectives on Multiagent Learning.” Artificial Intelligence 171 (2007): 382–391. 184Sandholm has been developing “hybrid” bots: Ganzfried, Sam, and Tuomas Sandholm. “Game Theory-Based Opponent Modeling in Large Imperfect-Information Games.” Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems 2 (2011): 533–540. 185professional

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, 205 Gambling Act, 71, 201 gambling laws, 20–21, 22, 71, 100, 198, 200, 201 gambling traits, traditional, 208 “Game of Poker, The” (Turing), 170 game theory, 137, 141, 148, 149, 151, 154, 155, 156, 160, 162, 163, 169, 178, 180, 181, 183, 208, 217 Gaming Act, 21 Gandy, Robin, 170 Ganzfried

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, 160, 161, 177, 181, 184, 188, 208 in rock-paper-scissors, 143, 178, 180 in soccer, 146, 147 and stock/financial markets, 161 See also game theory; perfect strategies “Optimum Strategy in Blackjack, The” (Baldwin, Cantey, Maisel, and McDermott), 37 order-routing algorithms, 115 Osipau, Andrei, 72 overestimating, 98 overlays bias, 56

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, 41, 46, 62, 63, 127, 217 Poisson, Siméon, 75 Poisson process, 75–76, 78 poker abstractions and, 212 analysis of the endgame in, 143 applying game theory to, 141, 148, 181, 183 bankroll management in, 144–145 basic options in, 138–139, 142 behaviors in, 191–192 coalitions in, 181–183 combined

Licence to be Bad

by Jonathan Aldred  · 5 Jun 2019  · 453pp  · 111,010 words

relevant to making decisions but of perfect and exhaustive powers of computation and logical reasoning. John von Neumann is usually seen as the father of game theory. Nash may have been a genius, but he was almost a mathematical minnow in comparison to von Neumann. DR STRANGELOVE AND THE KAISER’S

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’s major mathematical achievements. But by providing a more general approach to playing non-cooperative games, Nash’s equilibrium idea effectively superseded von Neumann’s game theory. And it casts light on a central aspect of what it means to be human – interdependence. Since our choices are interdependent, an individual’s

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in our thinking about human interaction. What happened next? NON-COOPERATION ABOUT NON-COOPERATION At first, nothing. Economists did not adopt game theory; a few mathematicians elaborated the mathematics of game theory as a project in pure mathematics; and RAND doggedly pursued a game-theoretic approach to military strategy, with few results of

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combined with some relatively minor criticisms from his PhD supervisor, led Nash, who struggled to cope with intellectual criticism, seriously to consider abandoning research in game theory altogether. By the end of the 1950s the root of his problems would become clear: diagnosed with paranoid schizophrenia, he would spend ever-longer periods

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s greatest intellectual accomplishments emerged in collaboration with others, and most of the (collaboratively written) Theory of Games and Economic Behavior was devoted to cooperative game theory, Nash was a loner. Indeed, he argued (in another path-breaking paper published just a year after his paper setting out the Nash equilibrium

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idea) that von Neumann’s cooperative game theory was redundant. All cooperative games, Nash argued, should be understood as in fact non-cooperative: the seemingly cooperative phase, involving players making agreements before

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social sciences from the 1960s onwards which ‘explain’ seemingly cooperative or altruistic behaviour as really non-cooperative and selfish underneath. Neither Nash nor his game theory did cooperation. From a comfortable post-Nobel Prize vantage point four decades later, Nash was phlegmatic about the initial rejection of his equilibrium concept by

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of von Neumann. In the mid-1950s he was busy with the development of the bomb and the computer. When he did have time for game theory, he reiterated his complaint that mainstream economic theory was mathematically primitive. In so doing he deeply antagonized many of the self-confessed ‘mathematical’ economists

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the academic audience likely to be most receptive to his game theory. Given von Neumann’s mathematical ambitions for social science, it was ironic that what finally propelled game theory beyond RAND and university maths departments was not maths but a story. Albert Tucker was John Nash’s PhD supervisor. In May 1950, just after persuading

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his wayward student not to abandon his PhD on game theory, Tucker was asked to talk about the new theory to a group of psychologists. Since his audience

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the blame for the damaging non-cooperation nurtured by this reasoning may seem to lie with John Nash’s equilibrium idea. But – although millions of students in social science, philosophy, law and biology are today introduced to game theory via the Prisoner’s Dilemma and its Nash equilibrium ‘solution’ – the Nash equilibrium idea

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must play the Prisoner’s Dilemma this way – and suffer the consequences, from longer jail terms to nuclear arms races. This follows directly from game theory’s assumption that rational human behaviour is non-cooperative and distrustful. Von Neumann could not imagine it otherwise: ‘It is just as foolish to complain

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, in contexts where the selfish, ruthless, calculating side of humanity is to the fore. But what contexts are those? From the 1960s onwards, as game theory began to creep out of its academic niche into wider discussion in social-science departments and beyond, it became clear that the biggest challenge to

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and educated to behave that way and experience confirms it generally makes life more liveable. Put another way, we escape the destructive consequences of following game theory’s prescription of ‘rational’ behaviour in the Prisoner’s Dilemma by rejecting this definition of ‘rational’. As Nobel Prize-winning economist Amartya Sen put

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secret society, hardly an association. It is formed by spontaneous generation.’12 In recent times, fans of Friedrich Hayek have invoked this aspect of game theory in an attempt to give mathematical credibility to Hayek’s idea that seemingly anarchic societies with little or no government can sustain themselves through ‘spontaneous

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order’. Although the obvious political home for game theory seems on the Right – relentless competition between selfish individuals, self-organizing societies with no need for government – thinkers on the Left have pressed it

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without any expectation of future interaction: they cannot assume they are playing a repeated game. Instead, they unconsciously rely on facts about human psychology which game theory ignores. It is, for instance, much easier to decide whether someone can be trusted if you meet them face to face. This is why,

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cooperative behaviour when people interact repeatedly over a fairly long period, what about one-off interactions? Here the gap between theory and reality remains: game theory predicts that people will not cooperate in one-off Prisoner’s Dilemma situations, yet they often do. Game theorists did not face up to this

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– the unsentimental expectation that people, individually and collectively, will behave more or less in their rational self-interest.13 Although, from the 1960s onwards, game theory began to influence everyday thinking, game theorists themselves were focusing on its limitations. In particular, they were becoming aware that in all too many contexts

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the point? Worse still, over the coming years it became clear that games with multiple Nash equilibria were not rare exceptions. They were ubiquitous. Game theory provided no guidance in these situations. And by the time the importance and ubiquity of this so-called multiplicity problem became clear, Nash was in

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and your opponent refuses the first offer they receive, then you know, from observing this refusal alone, that they are not following the rulebook of game theory, because backward induction implies they should accept the first offer. More generally, in real-life interactions beyond the game show, we frequently deal with

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honour. But Nash said he would have preferred to win the prize alone, because he really needed the money. He finished his speech by comparing game theory to string theory in physics, both subjects that researchers find intrinsically fascinating – so they like to pretend that both subjects are actually useful.16

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chess Grandmaster, then both of them might reasonably assume their opponent has a sophisticated knowledge of game theory. Such a defence of game theory is (a bit) less useless than it seems. On 5th December 1994, the day John Nash left America for Stockholm to collect his Nobel Prize, Vice-President Gore was announcing the ‘greatest

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auction ever’ – an auction of airwave frequency spectrum licences to be used by mobile phones. Auctions are a type of game, and this auction was carefully designed using the latest game theory. When the

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: it had received more than $7 billion in bids. The spectrum auctions, great revenue-raising successes for government, were hailed as a triumph of applied game theory. Here, at last, was a setting in which truly ‘rational’ players would interact – big corporations competing in an auction, each advised by a team

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which could be predicted and tweaked by the game theorists designing the auction, on behalf of the government. Or so it seemed. In reality, game theory did not provide the recipe for an ideal auction design to meet the government’s objectives, because the theory could not adjudicate between conflicting auction

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there was, the better. Dominant corporations were dominant, they argued, because they offered better products at lower prices, not because of anti-competitive practices. Game theory gave opponents of the Chicago view a new framework which took anti-competitive behaviour seriously, a framework which impressed regulators and courts because of its

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imperialistic forays into aspects of life beyond the scope of markets and prices, and hence beyond the scope of the traditional tools of economic analysis. Game theory provided a new toolkit for these economists, who saw themselves as social engineers designing institutions and mechanisms to produce desired social outcomes. In their

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how broken it seems, never dies. Many thinkers abandon the project, but new recruits revive the grand dreams. As one recent convert solemnly intoned, ‘game theory is a general lexicon that applies to all life forms. Strategic interaction neatly separates living from non-living entities and defines life itself.’20 These

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way, some subtleties have been lost. It is widely believed that cooperation is mostly for suckers and only the naïve rely on trust. In particular, game theory has been understood to prove, as a matter of irrefutable logic, that it is irrational to be altruistic, trustworthy or cooperative, even when the

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Yes, game theorists – especially in the early days of von Neumann, Nash and RAND – often assumed people are always selfish. But the circumstances under which game theory justifies or recommends selfishness are remarkably narrow. Nash’s equilibrium idea essentially implies that if everyone else is behaving selfishly, you should do so too

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behaving selfishly in the first place. And without this assumption, the explanation for why we get locked into non-cooperative situations disappears. Put another way, game theory says we will end up in a Nash equilibrium, but it does not explain which equilibrium – cooperative, non-cooperative or otherwise. It is a

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players are playing their equilibrium strategies, and more about whether we will reach that equilibrium in the first place: a question of history rather than game theory. (In the case of QWERTY, its convoluted form was precisely the point: it was invented to slow down typists in an era of mechanical

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concerns about fairness and responsibility typically look backwards: to the history of who did what, and why. This focus on consequences alone means that game theory must inevitably operate with a restricted, partial understanding of what it means to be human, an understanding which insists our future is always more important

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what it means. 4 The Government Enemy In the early 1950s John von Neumann and John Nash were not the only geniuses associated with the RAND Corporation. RAND was the incubator for another intellectual revolution, as significant as game theory but completely independent of it. And this time the genius behind it was a

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himself. The opponent should be thought of not as an individual but as a group, a collective. However, this interpretation poses a problem for game theory, because its players are assumed to have clear preferences between the alternatives they face. This assumption seems plausible for a hyper-rational individual but mysterious

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Morgenstern, and avowedly mathematical economists like Ken Arrow. In their work, humans were single-purpose robots, constantly calculating in order to maximize their ‘payoff’ (in game theory) or their ‘preference satisfaction’ (in Arrow and most mathematical economics). With this astonishingly limited representation of humanity in hand, there is no reason to restrict

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of Secretary of State Robert McNamara. Schelling received the Nobel Prize for economics in 2005 ‘for having enhanced our understanding of conflict and cooperation through game-theory analysis’. But this pigeonholing is too simplistic. Schelling did not see himself as a game theorist: his bemused response to the Nobel Prize announcement

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was ‘I must have been doing game theory without knowing it.’20 This reply was partly motivated by Schelling’s view of mathematics. Unlike most game theorists and RAND analysts, Schelling did not

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Keynesian economics: they thought that Keynes was a ‘charlatan’ and economics needed to be rebuilt from the ground up on the rigorous mathematical foundations of game theory. Uncertainty had to be captured in precise numbers. Von Neumann and Morgenstern had no time for Keynesian sentiments like ‘we simply do not know’.

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an arrogant, egotistical, flirtatious party animal. He knew how to get noticed in his academic work too. Ellsberg worked with Tom Schelling on applying game theory to nuclear strategy. Ellsberg provocatively titled one of his lectures ‘The Political Uses of Madness’ and argued that Hitler had been a successful blackmailer because

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H. (2009), The Bounds of Reason (Princeton: Princeton University Press). 21 For a rigorous development of the argument sketched below see F. Guala (2006), ‘Has Game Theory been Refuted?’, Journal of Philosophy, 103, 239–63. 22 https://www.aip.org/history-programs/niels-bohr-library/oral-histories/30665. 3. WEALTH BEATS JUSTICE

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principle, 57–63, 64–7, 137 predatory pricing, 57 principles for new relationship with, 251–61 privatization, 50, 54, 88, 93–4 rise of game theory, 40–41 Smith’s enlightened self-interest, 11 value of human life (‘statistical lives’), 141–5, 207 vocational role of, 260 see also behavioural economics

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, 68 as conditioned and limited by economics, 3, 10, 15, 43, 55, 60–61, 64–5, 179, 204–5, 218, 247 cooperative behaviour in game theory, 29, 30–32 core principles of current economic orthodoxy, 253 distinction between values and tastes, 136–8 economists’ language on virtuous behaviour, 112 inequality as

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free-riding behaviour European Commission, 96 Facebook UK, 99 fairness, 1, 149, 218, 228, 253 and Coase, 54, 55 and free-riding behaviour, 107 and game theory, 43 and incentives, 177, 179 and lucky geniuses, 221–3 and Posner’s wealth-maximization principle, 60, 61, 62 see also inequality family life, 127

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Business is to Increase Its Profits’ (article, 1970), 2, 152 Frost, Gerald, Antony Fisher: Champion of Liberty (2002), 7* Galbraith, John Kenneth, 242–3 game theory assumptions of ‘rational behaviour’, 18, 28, 29–32, 35–8, 41–3, 70, 124 Axelrod’s law of the instrument, 41 backward induction procedure, 36

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